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Register a new userUniversal Finite-Size Scaling around Topological Quantum Phase Transitions
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The critical point of a topological phase transition is described by a conformal field theory, where finite-size corrections to energy are uniquely related to its central charge. We investigate the finite-size scaling away from criticality and find a scaling function, which discriminates between phases with different topological indexes. This function appears to be universal for all five Altland-Zirnbauer symmetry classes with non-trivial topology in one spatial dimension. We obtain an analytic form of the scaling function and compare it with numerical results.
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A quantum tricritical point is shown to exists in coupled time-reversal symmetry (TRS) broken Majorana chains. The tricriticality separates topologically ordered, symmetry protected topological (SPT), and trivial phases of the system. Here we demonstrate that the breaking of the TRS manifests itself in an emergence of a new dimensionless scale, $g = alpha(xi) B sqrt{N}$, where $N$ is the system size, $B$ is a generic TRS breaking field, and $alpha(xi)$, with $alpha(0)equiv 1$, is a model-dependent function of the localization length, $xi$, of boundary Majorana zero modes at the tricriticality. This scale determines the scaling of the finite size corrections around the tricriticality, which are shown to be {it universal}, and independent of the nature of the breaking of the TRS. We show that the single variable scaling function, $f(w)$, $wpropto m N$, where $m$ is the excitation gap, that defines finite-size corrections to the ground state energy of the system around topological phase transition at $B=0$, becomes double-scaling, $f=f(w,g)$, at finite $B$. We realize TRS breaking through three different methods with completely different lattice details and find the same universal behavior of $f(w,g)$. In the critical regime, $m=0$, the function $f(0,g)$ is nonmonotonic, and reproduces the Ising conformal field theory scaling only in limits $g=0$ and $grightarrow infty$. The obtained result sets a scale of $N gg 1/(alpha B)^2$ for the system to reach the thermodynamic limit in the presence of the TRS breaking. We derive the effective low-energy theory describing the tricriticality and analytically find the asymptotic behavior of the finite-size scaling function. Our results show that the boundary entropy around the tricriticality is also a universal function of $g$ at $m=0$.
For systems with infinite-order phase transitions, in which an order parameter smoothly becomes nonzero, a new observable for finite-size scaling analysis is suggested. By construction this new observable has the favourable property of diverging at the critical point. Focussing on the example of the F-model we compare the analysis of this observable with that of another observable, which is also derived from the order parameter but does not diverge, as well as that of the associated susceptibility. We discuss the difficulties that arise in the finite-size scaling analysis of such systems. In particular we show that one may reach incorrect conclusions from large-system size extrapolations of observables that are not known to diverge at the critical point. Our work suggests that one should base finite-size scaling analyses for infinite-order phase transitions only on observables that are guaranteed to diverge.
I study the universal finite-size scaling function for the lowest gap of the quantum Ising chain with a one-parameter family of ``defect boundary conditions, which includes periodic, open, and antiperiodic boundary conditions as special cases. The universal behavior can be described by the Majorana fermion field theory in $1+1$ dimensions, with the mass proportional to the deviation from the critical point. Although the field theory appears to be symmetric with respect to the inversion of the mass (Kramers-Wannier duality), the actual gap is asymmetric, reflecting the spontaneous symmetry breaking in the ordered phase which leads to the two-fold ground-state degeneracy in the thermodynamic limit. The asymptotic ground-state degeneracy in the ordered phase is realized by (i) formation of a bound state at the defect (except for the periodic/antiperiodic boundary condition) and (ii) effective reversal of the fermion number parity in one of the sectors (except for the open boundary condition), resulting in a rather nontrivial crossover ``phase diagram in the space of the boundary condition (defect strength) and mass.
The interest in the topological properties of materials brings into question the problem of topological phase transitions. As a control parameter is varied, one may drive a system through phases with different topological properties. What is the nature of these transitions and how can we characterize them? The usual Landau approach, with the concept of an order parameter that is finite in a symmetry broken phase is not useful in this context. Topological transitions do not imply a change of symmetry and there is no obvious order parameter. A crucial observation is that they are associated with a diverging length that allows a scaling approach and to introduce critical exponents which define their universality classes. At zero temperature the critical exponents obey a quantum hyperscaling relation. We study finite size effects at topological transitions and show they exhibit universal behavior due to scaling. We discuss the possibility that they become discontinuous as a consequence of these effects and point out the relevance of our study for real systems.
We consider scaling of the entanglement entropy across a topological quantum phase transition in one dimension. The change of the topology manifests itself in a sub-leading term, which scales as $L^{-1/alpha}$ with the size of the subsystem $L$, here $alpha$ is the R{e}nyi index. This term reveals the universal scaling function $h_alpha(L/xi)$, where $xi$ is the correlation length, which is sensitive to the topological index.
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